Problem 9 - College Algebra

Question

Find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$x^{2}=12 y$$

Step-by-Step Solution

Answer

The focus of the parabola is at (0,3) and the directrix is the line y = -3. Graph of the parabola opens upwards with the vertex at the origin.

Step 1: Rearrange the Equation to Standard Form

First, rewrite the equation into standard form. The given equation is \(x^2 = 12y\). We need to compare it with the standard form \(x^2 = 4ay\), and by comparing, we can find that the value of \(4a\) is 12.

Step 2: Find the Value of 'a'

Find the value of 'a' by setting \(4a = 12\), the coefficient of y in the given equation. Solving this equation gives \(a = 3\).

Step 3: Find the Focus and Directrix

For a parabola in the form \(x^2 = 4ay\), the focus of the parabola is at the point (0, a), so in this case, (0,3). The equation of the directrix is \(y = -a\), so in this case, \(y = -3\).

Step 4: Graph the Parabola

Graph the equation using the function \(f(x) = x^2 /4a\). The vertex is at the origin (0,0), the focus is at (0,3) and the directrix is the line \(y = -3\). Make sure to also draw the axis of symmetry, which is the vertical line passing through the vertex and the focus.